Ground state factorization versus frustration in spin systems

1"Hamiltonian & Gaps", 7/9/2010

Ground state factorization versus frustration in spin

systemsGerardo Adesso

School of Mathematical SciencesUniversity of Nottingham

joint work with S. M. Giampaolo and F. Illuminati (University of Salerno)


Outline

• Spin systems and frustration• What we want to do and why• Theory of ground state factorization• Factorized solutions to frustration-free models• Frustration vs factorization and order• Summary and outlook


Quantum spin systems• N spin-1/2 particles

on regular lattices

– anisotropic interactions of arbitrary range

– arbitrary spatial dimension

– translationally invariant & PBC

– external field along z

H = 12Xi ;lJ rxSxi Sxl + J rySyi Syl + J rzSzi Szl ¡ hX

iSzi

( )r i l= -


Ground states

• No known exact analytical solution in general, except for a few simple subcases (Ising, XY,...) and now a wider class of models with nearest-neighbor interactions (see JE)

• Difficult to be determined even numerically, especially for high-dimensional lattices (2D, 3D, ...)

• Rich phenomenology: different magnetic orderings, critical points and quantum phase transitions

• Typically exhibit highly correlated quantum fluctuations, i.e., they are typically entangled


Frustration

AF ?

AF

AF

• Occurs when the ground state of the system cannot satisfy all the couplings

• Even richer phase diagram (high degeneracy), hence even harder to find ground states

• In frustrated systems a magnetic order does not freeze, which typically results in even more correlations

• At the root of statistically fascinating phenomena and exotic phases such as spin liquids and glasses

• Frustrated systems may play a crucial role to model high-Tc superconductivity and certain biological processes


Relevant questions• How to define natural signatures and measures of

(classical and/or quantum) frustration?

• More generally, is it possible to tune an external field so that a many-body model admits as exact ground state a completely factorized (“classical-like”) state?– This would be an instance of mean field becoming exact

• If yes, under which conditions? Does this possibility depend on the presence of frustration? In turn, does the fulfillment or not of this condition define a regime of weak versus strong frustration?


Ground state factorization

• Answer to the 2° question: YES! There can exist special points in the phase diagram of a spin system such that the ground state is exactly a completely uncorrelated tensor product of single-spin states: factorized ground state

• The “factorization point” is obtained for specific, finite values of the external magnetic field (dubbed factorizing field) which depend on the Hamiltonian parameters

• First devised by Kurmann, Thomas and Muller (1982) for 1-d Heisenberg chains with nearest neighbor antiferromagnetic interactions

ground = ⊗ ⊗ ⊗ ⊗ …


Motivations• Many-body condensed matter perspective

– To find exact particular solutions to non-exactly solvable models– To devise ansatz for perturbative analyses, DMRG, …

• Quantum information and technology perspective– For several applications (e.g. quantum state transfer, dense coding,

resource engineering for one-way quantum computation), both in the case of protocols relying on “natural” ground state entanglement for quantum communication (in which case factorization points should be avoided!), and for tasks which instead require a qubit register initialized in a product state

• Statistical perspective– To investigate the occurrence of “phase transitions in

entanglement” with no classical counterpart– For frustrated systems: to characterize the frustration-driven

transition between order (signaled by a factorized ground state) and disorder (landmarked by correlations in the ground state), thus achieving a quantitative handle on the frustration degree


History• Direct method (product-state ansatz) Analitic brute-force method, guess a product state and verify that it is the ground state

via the Schrödinger equation, becomes nontrivial for more complicate models …

– Kurmann et al. (1982): 1d Heisenberg, nearest neighbors– Hoeger et al. (1985); Rossignoli et al. (2008): 1d Heisenberg, arbitrary

interaction range– Dusuel & Vidal (2005): Fully connected Lipkin-Meshkov-Glick model– Giorgi (2009): Dimerized XY chains

• Numerical method (Monte Carlo simulations)Nightmarish for spatial dimensions bigger than two (never attempted !)

– Roscilde et al. (2004, 2005): 1d & 2d Heisenberg, nearest neighbors


Our method• Quantum informational approach

– Inspired by tools of entanglement theory– Fully analytic method– Requires no ansatz: the magnetic order, energy, and specific

form of the factorized ground state are obtained as a result of the method

– Encompasses previous findings and enables the identification of novel factorization points

– Provides self-contained necessary and sufficient conditions for ground state factorization (in absence of frustration) in terms of the Hamiltonian parameters

– Straightforwardly applied to cases with arbitrary range of the interactions and arbitrary spatial dimension (e.g. cubic Heisenberg lattices), and to systems with spatial anisotropy, etc.

S. M. Giampaolo, GA, F. Illuminati, Phys. Rev. Lett. 100, 197201 (2008); Phys. Rev. B 79, 224434 (2009)


The ingredients /1

• Under translational invariance, the ground state is completely factorized iff the entanglement between any spin and the block of all the remaining ones vanishes, i.e., if the marginal (linear) entropy of a generic spin, say on site k, is zero

• We have:so this factorization condition would depend on the magnetizations, which are indeed the objects one cannot compute in general models

½k

SL (½k) = 4Det½k =1¡ 4£hSxk i2+hSyk i2+hSzk i2¤


The ingredients /2

• A generic N-qubit state is factorized iff for any qubit k there exist a unique Hermitian, traceless, unitary operator Uk (which takes in general the form of a linear combination of the three Pauli matrices), whose action on qubit k leaves the global state unchanged (Giampaolo & Illuminati, 2007)

• We can define in general the “entanglement excitation energy” (EXE) associated to spin k as the increase in energy after perturbing the system, in its ground state, via this special local unitary Uk (Giampaolo et al., 2008)In formula:

• One can prove that, under translational invariance and under the hypothesis [H,Sa]≠0 (a=x,y,z), the ground state is completely factorized iff the entanglement excitation energy vanishes for any generic spin k

½k

Uk

¢ E (Uk) = hª jUykHUk jª i ¡ hª jH jª i


The ingredients /3

1. A factorized ground state must have vanishing local entropy2. A factorized ground state must have vanishing EXE3. The ground state must minimize the energy4. The Hamiltonian model H does not give rise to frustration

general theory of ground state factorization



Net interactions• All the results (form of the state, factorizing field, conditions

for ground state factorization) are only functions of the Hamiltonian coupling parameters and of lattice geometry factors, or more compactly, of the “net interactions”:

– Zr is the coordination number, i.e. the number of spins at a distance r from a given site

– the magnetic order is determined by¹ =minf J Fx ; J Ax ; J Fy ; J Ay g

¹ =

8>><>>:

J Fx ) Ferrom. order along x ;J Fy ) Ferrom. order along y ;J Ax ) Antiferrom. order along x ;J Ay ) Antiferrom. order along y .

S. M. Giampaolo, GA, F. Illuminati, Phys. Rev. B 79, 224434 (2009)

, ,1

, ,1

,,

1

( 1) ,

,

.

A r rx y r x y

r

F rx y r x y

r

A rx y r

rz

F

Z J

Z J

Z J

¥

=¥

=¥

=

= -

=

=

ååå

J

J

J


Results: Frustration-free



Heisenberg lattices

• The method is versatile and the result is totally general: the complexity is the same for any spatial dimension, one only needs to put the correct coordination numbers in the definition of the net interactions (e.g. for nearest neighbor models: Z=2 for chains, Z=4 for planes, Z=6 for cubic lattices)

F (x)

F (y)

AF (x)

AF (y)Kurmann‘82

1D nearest neighbor



Other applications

• Long-range and infinite-range models• Models with spatial anisotropy

• …

a) b)

c) d)



Frustrated systems

• We consider a subclass of the original Hamiltonian, comprising models with anisotropic antiferromagnetic (along x) interactions up to a maximum range rmax

• Frustration arises from the interplay between the couplings at different ranges• We focus on 1d systems (chains) of infinite length• For simplicity, we consider the interaction anisotropies independent on the

distance, but overall the couplings are rescaled by a range-dependent factor fr

• If all the fr’s beyond r=1 vanish, the system is not frustrated. Vice versa, if the fr’s are all equal, the system is fully frustrated.

AF

AF AF

H = Xi ;r · rmax

f r (J xSxi Sxi+r +J ySyi Syi+r +J zSzi Szi+r ) ¡ hXiSzi

S. M. Giampaolo, GA, F. Illuminati, Phys. Rev. Lett. 104, 207202 (2010)


Short-range systems

• Simplest case: rmax = 2 (nearest and next-nearest neighbors)

– We set f1 = 1, f2 ≡ f– The parameter f ∊[0,1] plays the role of a “frustration

degree”(a more general definition of frustration degree was given by Sen(De) et al., PRL 2008)

• Magnetic order of the ground statef < ½ standard antiferromagnet

f ≥ ½ dimerized antiferromagnet



Factorized ground states• We can determine in general the form of the candidate factorized state

and the factorizing field

• The nontrivial part is now in the verification steps– We find that for the candidate factorized state to be an eigenstate, a

necessary condition isJz = 0 (other possibilities lead to saturation instead of proper factorization)

– By decomposing the Hamiltonian into triplet terms, we can derive a sufficient condition for the candidate state to be the ground state, by testing whether its projection on three spins is the ground state of the triplet Hamiltonian

– For frustration-free, the factorized state was always the ground state

hf =12p J xJ y = (1¡ f )p J xJ y f < 1=2

f p J xJ y f ¸ 1=2



ground state factorization

Factorization vs frustration• From the triplet decomposition we find, analytically, that if the frustration

is weaker than a “critical” value, , then the ground state is factorized

• The actual “compatibility threshold” (i.e. the maximum frustration degree that allows ground state factorization) can be determined numerically by considering decompositions into blocks of more than three spins.

• Above this boundary the system admits a factorized eigenstate at h=hf, but this does not minimize energy and instead the ground state is entangled

ground state entanglement

12

x x y yc

x y

J J J Jff

J J- +

£ º +



Remarks

• Frustration naturally induces correlations which tend to suppress ground state factorization: for strong enough frustration it is not energetically favourable for the system to arrange in a factorized state (although a factorized state can exist in the higher-energy spectrum)

• At the factorizing field, we witness a first order quantum phase transition (level crossing) from a factorized to an entangled ground state when the frustration crosses the compatibility threshold

• Qualitative agreement with the results on the scaling of correlations (Sen(De) et al., PRL 2008) and on tensor network representability (Eisert et al., PRL 2010)

Factorized antiferromagnetic ground state

Factorized antiferromagnetic excited eigenstate

Factorized dimerized excited eigenstate

FRU

STRA

TIO

N

( h = hf )



Remarks

• Reversing the perspective, we can define the regime of weak frustration as the one compatible with ground state factorization, and the regime of strong frustration as the one where no factorization points are allowed.

• Ground state factorization implies a definite magnetic order, thus it is a precursor to a quantum phase transition, with critical field hc≥hf

• The regime of strong frustration is thus characterized by the fact that a magnetic order does not freeze even at zero temperature (in layman’s words, the ground state remains always entangled), in accordance with other criteria to assess the frustration degree (Ramirez, Balents, ...)

Factorized antiferromagnetic ground state

Factorized antiferromagnetic excited eigenstate

Factorized dimerized excited eigenstate

FRU

STRA

TIO

N

( h = hf )



Longer-range models• The same general features emerge by investigating frustrated

systems with interactions beyond next-nearest neighbors– Factorized eigenstates are only allowed for Jz=0

(this limitation could be relaxed in more general non-translationally-invariant models where the anisotropies depend individually on the distance)

– There is a compatibility threshold dividing the phase diagram into a region of weak frustration/order/ground state factorization and a region of strong frustration/disorder/ground state entanglement

Compatibility thresholds: Maximum value of thefrustration f as a function of the ratio J y=J x forwhich ground statefactorization pointsexist in frus-trated antiferromagnets with rmax = 4. The blackline stands for systems with f 2 = f , f 3 = f =2 andf 4 = f =3, the red line for f 2 = f , f 3 = f =2 andf 4 = f =4, and the blue line for f 2 = f , f 3 = f =p 2and f 4 = f =6.

rmax = 4



Infinite-range models

• To verify ground state factorization in fully connected models (rmax=∞), one should decompose the Hamiltonian in terms involving n→∞ spins, i.e., basically solve the Hamiltonian itself!

• A workaround is possible if the frustration coefficients fr follow a decreasing functional law with r and vanish in the limit r→∞

• In this case one can impose a cutoff and deal with decompositions into blocks of a finite number n of neighboring spins which are most effectively coupled

• Then one takes the limit n→∞. Numerically, this means that ground state factorization occurs if, for n large enough, the difference D between the minimum eigenvalue m of the n-spin Hamiltonian component and the energy associated to the candidate factorized state, vanishes asymptotically. If r’ is the cutoff range (such that n=2r’+1), then

PD(r0) = ¹ (r0)+ 14(J x +J y) r

l=1(¡ 1)lfl

fr



• Case fr=1/r2 (weak frustration)– The difference D(r’)=0 for any cutoff r’

Exact factorized ground state at

• Case fr=1/r (medium frustration)– The difference D(r’) seems to converge

to 0 (more numerics needed) Conjectured factorized

ground state at

• Case fr=1/√r (strong frustration)– The difference D(r’) converges

to a finite value No factorized ground state !

Results for infinite range

hf = (¼2=12)p J x J y

hf = ln(2)p J x J yDD

_



Summary• We approached the problem of finding exact factorized ground state

solutions to general cooperative spin models• We devised a method to identify and fully characterize such solutions

thanks to some tools borrowed from quantum information theory• In frustration-free systems, necessary and sufficient conditions are derived

and several novel factorized exact solutions are straightforwardly obtained for various translationally invariant models, with interactions of arbitrary range, and arbitrary lattice spatial dimension

• In frustrated systems, a universal behaviour emerged in which frustration and ground state factorization are competing phenomena, the former inducing correlations and disorder, and the latter relying on ordered, uncorrelated magnetic arrangements. Notably short-range as well as infinite-range (weakly) frustrated antiferromagnetic models have been shown to admit exact factorized solutions

• Ground state factorization is an effective tool to probe quantitatively frustrated quantum systems. The possibility vs impossibility of having a classical-like ground state at a given value of the magnetic field defines the regimes of weak vs strong frustration


Discussion and outlook• In this talk we only considered spin-1/2 systems, however due to a

theorem by Kurmann et al. (1982), any spin-S (S> ½) Hamiltonian which is of the same form as a spin-1/2 Hamiltonian that admits a factorized ground state at h=hf, will also admit a factorized ground state at the same value of the field: greater scope of our results

• For a generic model (frustrated or not), the factorizing field (when it exists) is a precursor to the critical field associated to a quantum phase transition (where the external field is the order parameter), i.e. hf≤hc

• A fascinating perspective is the investigation of the ground state entanglement structure near a factorization point: it is conjectured that entanglement undergoes a global reshuffling and can change its typology (demonstrated in the XY and XXZ models, Amico et al. 2007): an “entanglement phase transition” with no classical counterpart, and signaled by a diverging range of pairwise entanglement

• More perspectives– Generalize the method: relax translational invariance, identify exactly

dimerized solutions, etc. – Area laws for frustrated systems?– Define a measure of frustration, able to distinguish quantum from classical– ...


Thank you?

GAP

Ground state factorization versus frustration in spin systems

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